Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations: (T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F. Prove tat S with these operations is a vector space over F. (b) In S, we have elements fi : V -> F...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
just part c,d, and e please!! Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe (C[0, 1], || |co) : 0 <f() < 1}. Show that X is complete but Y is not complete .
*14. Let A be an n x n matrix. Define f:R" R by f(x) = Ax.x = x'AX. (a) Show that f is differentiable and Df (a)h = Aah + Ah a. (b) Deduce that when A is symmetric, Df(a)h = 2Aa . h. 15. Let a € R", 8 >0, and suppose f: B(a, 8) - R is differentiable at a. Suppose f(a) f(x)
Problem 5. Let W and U be finite-dimensional vector spaces, and let T : W > W and S : W -> U be linear transformations. Prove that if rank(S o T) L W W such that S o T = So L. = rank(S), then there exists an isomorphism (,.. . , Vk) is a basis of ker(T), and let (w1, ., wr) is a basis of im(T) nker(S) if 1 ik Hint: Let B (vi,... , Vk,...,vj,) be...
Let H ≤ G, and g ∈ G, Define f : H → gHg^−1 by h → ghg^−1 . Show that f is an isomorphism. Hence |gHg^−1 | = |H|.
Could someone pls explain question 9 (e)? 9. Consider the set of matrices F = a) Show that AB BA for all A, B E F b) Show that every A E F\ {0} is invertible and compute A-. c) Show that F is a field d) Show that F can be identified with C e) What form of matrix in F corresponds to the modđulus-argument form of a complex number Comment on the geometric significance. Solution a) Let A...
(a) Find (22,8,P) and nonempty sets (An)>1 C 8 such that P (liminf An) <liminf (P(Ar)) < limsup(P (A)) <P (limsup An). (b) Given A, B,(A.)21,(B.).>1 C , either prove the following statements or find counterexamples to them. i. limsupA, U limsupB = limsupA, UB.. ii. liminf Anu liminfB = liminfAUB iii. limsupA, nlimsupB. = limsupA, B. iv. liminfa, nliminfB = limin A, B, (e) Prove that probability spaces have the property of continuity from above. (We proved continuity from...